Ideas for Undergraduate Mathematics Projects

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Ideas for Undergraduate Mathematics Projects

• Kevin Peterson
• Columbus State University

Project NExT
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CRC PublishingISBN 0-8493-8524-5
Motzkin-Dirac conjecture Any set on n
noncollinear points in the plane determines at
least n/2 ordinary lines (ngt13).
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Arrangement with 11 points
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Arrangement with 11 points
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AK PetersISBN 1-56881-111-X
Proposed by Erdös, Faudre, Pach and
Spencer Every triangle free graph on 5n
vertices can be made biparite by removing at
most n2 edges
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An example of a graph with 10 vertices
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MAA SpectrumISBN 0-88385-502-X
In 1907 R.D. Carmichael conjectured ?(x)n
never has exactly one solution for x.
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?(5)4 ?(8)4 ?(9)4 ?(10)4 ?(12)4
An example
?(1)1 ?(2)1
?(6)2 ?(4)2
Interesting result of Grosswald (1973)
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If such an n exists then it is divisible by
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Buffons Needle Problem

• Drop needle on a lined sheet of paper
• Needle is L units long
• Lines are D units apart (with DgtL)
• Find P(needle lands on a line)

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The Set Up
y
L/2 Sin ?
?
D
Needle Center
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The Solution

• Needle crosses line when y lt L/2 Sin ?

P(crossing)(area under curve)/(area of
rectangle) 2L/?D
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Generalizations
Let L gt D and let n number of lines crossed
by needle then calculate 1. P( exactly 1
crossing) 2. P(n2) 3. P(ngta) for a fixed a
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GeneralizationsDifferent floors?
Ask the same questions
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AK PetersISBN 1-56881-115-2
What properties must a graph have to guarantee it
to have a Hamiltonian Cycle?
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Better question for a student
Prove that the graph obtained from the
edge skeleton of an n-cube always has
a Hamiltonian Cycle.
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Proof for n3
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Undergraduate ResearchArchive
Located at www.maths.abdn.ac.uk/maths/department/
services/lms/
Give over 160 different research projects in
thefollowing format
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Example 1
Aspects of Random Neural Nets SOURCE Dr
Kevin Bowman AREA computing /modeling KEYS
neural networks, chaos LEVEL final year
undergraduate LENGTH 1/7 of year. 50-100 page
report
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PREREQ an initial programming course HISTORY
The project has actually been used in this form
DESCRIPTION Design and implementation of
Boolean networks, each node having up to 10
inputs. Investigation of chaotic behavior for
systems with 50/200/500 nodes 1. Report on and
implement the work in 2. References 1
Martland D, Dynamic behaviour of Boolean Networks
in Neural Computing Architectures ed
Alexsander I (217-238) 2 Sompolinsky H et al
(1988) Chaos in Random Neural Networks Phys Rev
Letters (61) 259-262
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Example 2
TITLE Square functions and curves SOURC
T.P.McDonough, Department of Mathematics,
University of Wales, Aberystwyth. AREA
Analysis KEYS periodic functions,
approximating functions, Fourier series.
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DESCRIPTION The graphs of two simple periodic
functions are sketched -- These functions may
also be approximated by functions such as a1
sin kx a2 sin 2kx a3 sin 3kx by choosing the
constants a1, a2, a3 and k appropriately.
Make a number of choices and quantify how well
your approximations fit the functions.
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Academic PressISBN 0-12-519260-6

• Combinatorial Algorithms
• Wilf and Nijenhuis

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Combinatorial Algorithms

• Contains more than 50 algorithms in pseudo-code,
  with explanations including
• Give all subsets of a finite set
• Give all k-subsets of a finite set
• Give next partition of n (in lexorder)
• Give next permutation of 1- n (in lexorder)
• Find a Hamiltonian cycle in a given graph